hjkMpfr {Rmpfr}

Hooke-Jeeves Derivative-Free Minimization R (working for MPFR)

Description

An implementation of the Hooke-Jeeves algorithm for derivative-free optimization.

This is a slight adaption hjk() from package dfoptim.

Usage

hjkMpfr(par, fn, control = list(), ...)

Arguments

par

Starting vector of parameter values. The initial vector may lie on the boundary. If lower[i]=upper[i] for some i, the i-th component of the solution vector will simply be kept fixed.

fn

Nonlinear objective function that is to be optimized. A scalar function that takes a real vector as argument and returns a scalar that is the value of the function at that point.

control

list of control parameters. See Details for more information.

...

Additional arguments passed to fn.

Details

Argument control is a list specifing changes to default values of algorithm control parameters. Note that parameter names may be abbreviated as long as they are unique.

The list items are as follows:

tol

Convergence tolerance. Iteration is terminated when the step length of the main loop becomes smaller than tol. This does not imply that the optimum is found with the same accuracy. Default is 1.e-06.

maxfeval

Maximum number of objective function evaluations allowed. Default is Inf, that is no restriction at all.

maximize

A logical indicating whether the objective function is to be maximized (TRUE) or minimized (FALSE). Default is FALSE.

target

A real number restricting the absolute function value. The procedure stops if this value is exceeded. Default is Inf, that is no restriction.

info

A logical variable indicating whether the step number, number of function calls, best function value, and the first component of the solution vector will be printed to the console. Default is FALSE.

If the minimization process threatens to go into an infinite loop, set either maxfeval or target.

Value

A list with the following components:

par

Best estimate of the parameter vector found by the algorithm.

value

value of the objective function at termination.

convergence

indicates convergence (TRUE) or not (FALSE).

feval

number of times the objective fn was evaluated.

niter

number of iterations (“steps”) in the main loop.

Note

This algorithm is based on the Matlab code of Prof. C. T. Kelley, given in his book “Iterative methods for optimization”. It has been implemented for package dfoptim with the permission of Prof. Kelley.

This version does not (yet) implement a cache for storing function values that have already been computed as searching the cache makes it slower.

Author(s)

Hans W Borchers [e-mail]; for Rmpfr: John Nash, June 2012. Modifications by Martin Maechler.

References

C.T. Kelley (1999), Iterative Methods for Optimization, SIAM.

Quarteroni, Sacco, and Saleri (2007), Numerical Mathematics, Springer.

See Also

Standard R's optim; optimizeR provides one-dimensional minimization methods that work with mpfr-class numbers.

Examples

## simple smooth example:
ff <- function(x) sum((x - c(2:4))^2)
str(rr <- hjkMpfr(rep(mpfr(0,128), 3), ff, control=list(info=TRUE)))

doX <- Rmpfr:::doExtras(); cat("doExtras: ", doX, "\n") # slow parts only if(doX)

## Hooke-Jeeves solves high-dim. Rosenbrock function  {but slowly!}
rosenbrock <- function(x) {
    n <- length(x)
    sum (100*((x1 <- x[1:(n-1)])^2 - x[2:n])^2 + (x1 - 1)^2)
}
par0 <- rep(0, 10)
str(rb.db <- hjkMpfr(rep(0, 10), rosenbrock, control=list(info=TRUE)))
if(doX) {
## rosenbrook() is quite slow with mpfr-numbers:
str(rb.M. <- hjkMpfr(mpfr(numeric(10), prec=128), rosenbrock,
                     control = list(tol = 1e-8, info=TRUE)))
}


##  Hooke-Jeeves does not work well on non-smooth functions
nsf <- function(x) {
  f1 <- x[1]^2 + x[2]^2
  f2 <- x[1]^2 + x[2]^2 + 10 * (-4*x[1] - x[2] + 4)
  f3 <- x[1]^2 + x[2]^2 + 10 * (-x[1] - 2*x[2] + 6)
  max(f1, f2, f3)
}
par0 <- c(1, 1) # true min 7.2 at (1.2, 2.4)
h.d <- hjkMpfr(par0,            nsf) # fmin=8 at xmin=(2,2)
if(doX) {
## and this is not at all better (but slower!)
h.M <- hjkMpfr(mpfr(c(1,1), 128), nsf, control = list(tol = 1e-15))
}
## --> demo(hjkMpfr) # -> Fletcher's chebyquad function m = n -- residuals